Bilevel Polynomial Programs and Semidefinite Relaxation Methods

TL;DR

This paper introduces a novel approach for solving bilevel polynomial programs by reformulating them as semi-infinite polynomial programs and applying semidefinite relaxation techniques, demonstrating convergence and efficiency through numerical experiments.

Contribution

It develops a new reformulation of bilevel polynomial programs as semi-infinite polynomial programs and proposes semidefinite relaxation methods with proven convergence for simple cases.

Findings

Convergence to global solutions for simple BPPs.

Efficient algorithms demonstrated through numerical experiments.

Reformulation as semi-infinite polynomial programs enhances solvability.

Abstract

A bilevel program is an optimization problem whose constraints involve another optimization problem. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as semi-infinite polynomial programs (SIPPs), using Fritz John conditions and Jacobian representations. Combining the exchange technique and Lasserre type semidefinite relaxations, we propose numerical methods for solving both simple and general BPPs. For simple BPPs, we prove the convergence to global optimal solutions. Numerical experiments are presented to show the efficiency of proposed algorithms.